Topological order in the color-flavor locked phase of - - PowerPoint PPT Presentation

Topological order in the color-flavor locked phase of (3+1)-dimensional U(N) gauge-Higgs system

Ryo Yokokura (KEK)

• 2019. 8. 19

Strings and Fields 2019 @ YITP Based on Y. Hidaka, Y Hirono, M. Nitta, Y. Tanizaki, RY, 1903.06389

Overview of this talk

v v v

= exp 2πik 3k + 1 ∈ Z3k+1

(for N = 3)

Color-flavor locked phase of a U(N) gauge theory with N-Higgs fields is topologically ordered if the Higgs fields have non-trivial U(1) charge k.

• Non-Abelian vortex and Wilson loop have a ZNk+1 fractional linking

phase.

• There are ZNk+1 1- and 2-form symmetries, and both of them are

spontaneously broken.

1 Introduction 2 Topological order in Abelian Higgs model 3 Topological order in CFL phase of U(N) gauge-Higgs system

Higgs phase of gauge theories

• Massive gauge fields
• Some Nambu–Goldstone (NG) bosons are eaten
• Admitting extended objects e.g. vortex
• Vortex in many contexts:
• magnetic vortex in superconductor (SC),
• (local) cosmic strings in cosmology

Higgs phases can be further classified by “topological order”

Topological order [Wen ’89, ’91]

v v v

= exp 2πi k ∈ Zk

Classification of phases by topology of non-local order parameters

• Order parameters: Wilson loop, vortex surface,...
• New classification e.g. SC ̸= charge 1 Abelian Higgs

A characterization of the topologically ordered phase

• 1. Non-zero VEV of non-local order parameters
• 2. Fractional linking phases between non-local operators

Is topological ordered phase related to symmetry breaking?

p-form global symmetries & their breaking

[Banks & Seiberg ’10; Kapustin & Seiberg ’14; Gaiotto et al. 14]

= e

2πi k

v v v

= e

2πi k

v v

Symmetry under transf. of p-dim. extended objects

• Charged objects: Wilson loop, vortex surface, ...

→ topological order parameters can be charged objects

• Example: phase rotation of a Wilson loop (1-form sym.)
• Symmetry breaking: non-zero VEV of charged objects

Topological order & p-form symmetry

[Banks & Seiberg ’10; Kapustin & Seiberg ’14; Gaiotto et al. 14]

Topologically ordered phase can be characterized by

• 1. p-form symmetry and their breaking
• 2. Fractional linking phases between non-local operators

In this talk, we consider the possibility of topologicaly ordered phase in 3 + 1 dim. non-Abelian gauge theories in order to understand phases of non-Abelian gauge theories.

Topological order in Abelian Higgs model

Review based on Hansson, et al. 04,; Banks & Seiberg ’10; Seiberg & Kapustin ’14; Gaiotto, et al. ’14

Message

v v v

= exp 2πi k ∈ Zk

Abelian Higgs models can be classified by topological order.

• Order parameters are non-local: Wilson loop & Vortex surface.
• Fractional linking phase of extended objects
• Symmetry breaking: 1- & 2-form symmetries
• E.g. (s-wave) superconductor ̸= charge 1 Higgs model

In the following, we consider the low energy limit with no local excitations

BF-theory as a dual of Abelian Higgs model

[Horowitz ’89; Blau & Tompson ’89]

Low energy (≪ v) limit of Abelian Higgs model is given by St¨ uckelberg action

dual

← → BF-action

v2 2

∫ |dχ − kA|2

dual

← → ik 2π ∫ B ∧ dA or

ik 2π

∫ d4xϵmnpqBmn∂pAq

• k: charge of Higgs field
• A: U(1) 1-form gauge field
• NG boson χ

dual

← → 2-form gauge field B

• B can be coupled with a magnetic vortex

∫

S B (S: worldsheet of vortex)

• No local excitations in both theories

How about observables and correlation functions in BF-theory?

Observables and correlation functions in BF-theory

[Horowitz & Srednicki ’89; Oda & Yahikozawa ’89]

Observables: non-local, gauge invariant

C

Wilson loop W(C) = ei

R

C A

Surface of vortex V (S) = ei

R

S B

v v v

S

at a time slice

Correlation functions are topological: linking number link (C, S) & Zk

• ⟨W(C)V (S)⟩

= e

2πi k

link (C,S) ⟨W(C)⟩ = e

2πi k

link (C,S) ⟨V (S)⟩

= exp( 2πi

k link (C, S))

• ⟨W(C)⟩ = ⟨V (S)⟩ = 1

v v v

1- and 2-form symmetries exist, and are broken spontaneously.

1- & 2-form symmetries [Kapustin & Seiberg ’14; Gaiotto, et al. ’14]

= e

2πi k

v v v

= e

2πi k

v v

Symmetry under transf. of 1- & 2-dim. objects Group Object Generator Transf. 1-form Zk W(C) V (S) ⟨V (S)W(C)⟩ = e

2πi k ⟨W(C)⟩

2-form Zk V (S) W(C) ⟨W(C)V (S)⟩ = e

2πi k ⟨V (S)⟩

• Symmetry breaking: ⟨W(C)⟩ ̸= 0, ⟨V (S)⟩ ̸= 0 (at large distance limit)

Notes

• ordinary symmetry is 0-form symmetry for particles
• BF -action is invariant under 1- & 2-form transf. (up to 2πZ).

Topological order in Abelian Higgs model [Hansson et al. ’04]

v v v

= exp 2πi k ∈ Zk

• 1. Existence of fractional linking phase

⟨W(C)V (S)⟩ = exp ( 2πi

k link (C, S)

)

• 2. 1- & 2-form symmetries are broken spontaneously.
• Order parameters = charged objects W(C) & V (S)
• Non-zero VEVs of order parameters: ⟨W(C)⟩ = ⟨V (S)⟩ = 1

E.g. superconductor (Z2) ̸= charge 1 Higgs system (Z1 = 1)

Summary of Abelian Higgs model

v v v

= exp 2πi k ∈ Zk

Abelian Higgs models can be classified by topological order.

• Order parameters are Wilson loop & Vortex surface.
• Fractional linking phase of extended objects
• Symmetry breaking: 1- & 2-form symmetries

Topological order in CFL phase of U(N) gauge-Higgs system

• Y. Hidaka, Y Hirono, M. Nitta, Y. Tanizaki, RY,

1903.06389

Topological order in non-Abelian Higgs models?

• Many symmetry breaking patterns
• Some of gauge fields are massive, others are not.
• Some NG boson may not be eaten.

(cf. no topological order in CFL of QCD [Hirono & Tanizaki ’18])

• Hints from Abelian case:
• No local excitations in low-energy limit
• Existence of extended objects

The hints suggest: there may be a topologically ordered phase in a U(N) gauge theory with N-flavor Higgs fields

U(N) gauge theory with N-flavor Higgs

[Hanany & Tong ’03; Auzzi et al. ’03; Gorsky et al. 04]

Action ∫ 1 2g2

1

tr |F|2 + 1 2g2

2

| tr F|2 + |dΦ − iAΦ − ik tr (A)Φ|2 + V (Φ)

• Higgs fields: N(color) × N(flavor) matrix Φ = (Φcf)
• Transf. law Φ → (det Ucol)k Ucol Φ U T

flav Ucol ∈ U(N)col, Uflav ∈ SU(N)flav/(ZN)flav

• A: U(N) 1-form gauge field

How about Higgs phase?

Higgs phase with non-Abelian vortex

[Hanany & Tong ’03; Auzzi et al. ’03; Gorsky et al. 04]

• 1. There is a Higgs phase with no massless excitation

(# of gauge fields = # of Higgs fields)

• VEV of Higgs can be diagonized ⟨Φ⟩ = v1N×N
• Color-flavor locked (CFL) phase:

simultaneous color-flavor transf. remains

(cf. QCD case [Alford et al. ’98])

• 2. CFL phase Admits non-Abelian vortices Φ ∼ diag (eiθ, 1, ..., 1)
• Fractional (1/N) magnetic flux

We will show that this CFL phase can be a topologically ordered phase.

diag (eiθ, 1, ..., 1) = eiθ/N diag (ei(N−1)θ/N, e−iθ/N, ..., e−iθ/N) ∼ eiθ/N1N×N

How to see topological order?

Conditions for the topological ordered phase

• 1. Existence of fractional linking phase,
• 2. Spontaneously broken 1- and 2-form symmetries

Procedure: similar to the case of Abelian Higgs model

• 1. Dual theory:

∫ |dΦ − iAΦ − ik tr (A)Φ|2

dual

← →

i 2πKiA

∫ bi ∧ daA

• 2. Fractional linking phase
• 3. 1- & 2-form symmetry breaking

In this talk, we consider N = 3 case for simplicity.

Low energy limit in CFL phase

Action is simplified in Abelian gauge: (cf. [’t Hooft ’81]) St¨ uckelberg action

v2 2

∫ |dφi − KiAaA|2, KiA =  

k + 1 k k k k + 1 k k k k + 1

 

• Abelian gauge: Φ =

1 √ 2v diag (eiϕ1, eiϕ2, eiϕ3) ∈ U(3)

• aA: Cartan of U(3) gauge field
• KiA: matrix of charges (det(KiA) ̸= 0)

|dϕi − KiAaA|2 = |dϕ1 − (k + 1)a1 − ka2 − ka3|2 + |dϕ2 − ka1 − (k + 1)a2 − ka3|2 +|dϕ3 − ka1 − ka2 − (k + 1)a3|2

Dual BF-type theory

The action i 2π KiA ∫ bi ∧ daA NG boson φi

dual

← → 2-form bi

Deriavtion

• 1. Original action v2

2

∫ |dφi − KiAaA|2

• 2. First order action by adding 3-form H3i

1 8π2v2

∫ |H3i|2 +

i 2π

∫ H3i ∧ (dφi − KiAaA)

• 3. Eliminating φi by EOM: dH3i = 0 → H3i = dbi
• 4. Take low-energy limit (v → ∞)

Observables in

i 2πKiA

∫ bi ∧ daA

W(C)

ex

v v v

V (S)

• Wilson loop: W(C) = 1

N tr Pei ∫

C A = 1

N N

∑

A=1

ei

∫

C aA

• Vortex surface operator: Vi(S) = ei

∫

S bi

Observables depends on only topology

(No local DOF: daA = 0, dbi = 0 by EOM)

CFL phase is topologically ordered!

v v v

= exp 2πik 3k + 1 ∈ Z3k+1

• 1. Z3k+1 fractional phase in correlation function

⟨W(C)Vi(S)⟩ = exp ( 2πik 3k + 1 link (C, S) )

• 2. Z3k+1 1- & 2-form symmetries & breaking
• ⟨W(C)Vi(S)⟩ = exp

(

2πik 3k+1 link (C, S)

) ⟨W(C)⟩

• ⟨W(C)Vi(S)⟩ = exp

(

2πik 3k+1 link (C, S)

) ⟨V (S)⟩

• Non-zero VEVs ⟨W(C)⟩ = ⟨Vi(S)⟩ = 1

Some comments

v v v

= exp 2πik 3k + 1 ∈ Z3k+1

• U(1) charge k in Φ → (det Ucol)k Ucol Φ U T

flav is important for the

existence of topological order.

• Non-Abelian vortices in the previous research [Gorsky, et al. 04]: k = 0

case (Z1). No nontrivial topological order.

• Possible fractional phase is restricted by color N:

e.g. Z4, Z7, Z11, ... are allowed for N = 3 case.

Summary

CFL phase of a U(N) gauge theory with N-Higgs fields is topologically

• rdered if the Higgs fields have non-trivial U(1) charge k.
• Non-Abelian vortex and Wilson loop has ZNk+1 fractional linking

phase.

• There are ZNk+1 1- and 2-form symmetries, and both of them are

spontaneously broken. Future work: more general gauge group, numerical solutions of charge k non-Abelian vortex, adding θ-term,...

Appendix

Derivation of correlation function in BF-theory I

We begin with the following correlation function [Chen, et al. ’15] ⟨W(C)V (S)⟩ = N ∫ DADBe

ik 2π

∫ B∧dA+i ∫ A∧J3(C)+i ∫ B∧J2(S)

where J3(C) and J2(S) are delta function forms. They can be rewritten as J3(C) = J3(∂S(C)) = dJ2(S(C)), J2(S) = J2(∂V(S)) = −dJ1(V(S)), where S(C) and V(S) are boundaries of C and S: ∂S(C) = C, ∂V(S) = S

Derivation of correlation function in BF-theory II

By the redefinition A → A + 2π

k J1(V(S)), we obtain

⟨V (S)W(C)⟩ = N ∫ DADB e

ik 2π

∫ B∧d(A− 2π

k J1(V(S)))+i

∫ A∧J3(C)

= N ∫ DADB e

ik 2π

∫ B∧dA+i ∫ A∧J3(C)+ 2πi

k

∫ J1(V(S))∧J3(C)

= e− 2πi

k

link (C,S) ⟨W(C)⟩ .

By the redefinition B → B − 2π

k J2(S(C)), we also obtain

⟨V (S)W(C)⟩ = N ∫ DADB e

ik 2π

∫ (B+ 2π

k J2(S(C)))∧dA+i

∫ B∧J2(S)

= N ∫ DADB e

ik 2π

∫ B∧dA+i ∫ B∧J2(S)− 2πi

k

∫ J2(S(C))∧J2(S)

= e− 2πi

k

link (C,S) ⟨V (S)⟩

Derivation of correlation function in BF-theory III

We can similarly show ⟨V (S)⟩ = 1 and ⟨W(C)⟩ = 1 as follows: ⟨V (S)⟩ =N ∫ DADB e

ik 2π

∫ B∧d(A− 2π

k J1(V(S)))

=N ∫ DADB e

ik 2π

∫ B∧dA = 1,

and ⟨W(C)⟩ =N ∫ DADBe

ik 2π

∫ (B+ 2π

k J2(S(C)))∧dA

=N ∫ DADBe

ik 2π

∫ B∧dA = 1.

By using there relations, we also obtain ⟨W(C)V (S)⟩ = e− 2πi

k

link (C,S).